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On Monday, Rubio put out a statement saying he would not seek and does not want to be considered for the position

(CNN)Marco Rubio on Tuesday declined to offer Donald Trump a formal endorsement and said he's not interested in being Trump's vice president, although will support the GOP nominee in accordance with a pledge he made during the primary fight.

"I stand by the things that I said" during the campaign, the Florida senator told CNN's Jake Tapper on "The Lead," referring to his past criticisms of the presumptive nominee, including calling him a "con man" and saying he shouldn't be given access to nuclear codes. "But I'm not going to sit here right now and become his chief critic over the next six months, because he deserves the opportunity to go forward and make his argument and try to win."

But Rubio stopped short of offering Trump his clear backing.

"I signed a pledge, put my name on it, and said I would support the Republican nominee and that's what I intend to do," he said.

Rubio's comments come as Trump's sudden status as the party's presumptive presidential nominee have the campaign transitioning from a fierce primary battle to preparing for the general election, unifying support from Republican leaders and beginning the process of selecting a running mate.

Rubio: Trump would be better served by VP who embraces the things he stands for. "That's certainly not me." https://t.co/3G2frjJvzN

Rubio on Tuesday repeated his statement that he did not want to be considered as Trump's running mate.

"He's the presumptive nominee at this point but he'd be best served by having someone, not just, by the way, a vice presidential nominee, but active surrogates who agree with him on his issues," Rubio told Tapper. "My differences with Donald, both my reservations about his campaign and my policy differences with him are well-documented."

Cruz could still be a headache for Trump

One other Trump primary challenger who does not appear ready to endorse the real estate mogul's is Texas Sen. Ted Cruz, who returned to the Senate, where he refused to rule out himself of the Republican race for good.

Cruz still controls a large bloc of delegates to the GOP convention and has the support of many movement conservatives, and declined to say whether they should jojn with Trump, as Republican Party leaders have been urging, or fight the presumptive nominee.

Cruz did dismiss a third-party challenge to Trump and Hillary Clinton, but suggested he could be wooed back in to the Republican race, despite dropping out one week ago in Indiana.

There you have it -- Ted Cruz has been offered two chances to back Donald Trump. He passed both times.

"Listen we have suspended the campaign because I can see no viable path to victory. Of course if that changed we would reconsider things," Cruz said Tuesday, to a Senate packed with reporters and TV cameras. "We've withdrawn from the campaign and it is in the hands of the voters. If circumstances change we will always assess changed circumstances."

Rubio as his running mate

While Rubio said on CNN Tuesday that he's not interested and that no discussions about the Trump ticket have reached him, a Trump campaign source said a longtime Rubio adviser discussed the idea of a Rubio running mate with party members at the Republican National Committee spring meeting in Hollywood, Florida, last month, pitching it as a possible way to attract Hispanic voters to Trump's campaign.

"This was RNC members, including myself, discussing different options," Fortuno said. The conversations were based on "who could strengthen the ticket."

Fortuno mentioned that he was not aware whether Rubio knew that his name was coming up as a potential running mate for Trump at the meetings. But the former governor noted there were approximately 160 members of the party engaged in these conversations.

Asked whether Trump and Rubio can overcome their differences, Fortuno was dubious. "Easier said than done, of course."

But a Trump campaign source painted a different picture, telling CNN that Rubio's team pitched the senator "hard," saying "he would make a great VP and he wants it."

When asked about Rubio's statement, the Trump campaign source quipped, "plausible deniability," noting people close to the senator had discussed the idea with party members.

A Republican who has advised Rubio for years confirmed the discussion took place, but said that "some things would have to change" for any formal alliance to emerge. "I think Marco has moved on," he added.

Before he left the Republican primary race, Rubio called Trump a "con artist" whose business empire was built on inherited money and foreign labor.

But that final volley of attacks, for which he later apologized, failed to trip up Trump, who won Rubio's home state of Florida by more than 18 points March 15.

Rubio dropped out that night. Returning to finish out his Senate term, Rubio has said he was "not going to be anybody's vice president."